I don't say that investigating whether Akai is dead or not is not important at all. But the action Bourbon should do now, if he is really loyal to BO, is to uncover Kir. And as I said, from that he could investigate about Akai, either by doing some manipulation on Kir, or capturing Kir and torturing (Kir may not have any motivation to keep the secret about Akai's fake death, since the fake death is initially to protect Kir's cover without losing Akai, I assume)